On generalizing Takahashi’s nonconvex minimization theorem

We present a simple proof of vectorial Takahashi’s nonconvex minimization theorem based on Gopfert, Tammer and Zalinescu [A. Gopfert, C. Tammer, C. Zalinescu, On the vectorial Ekeland’s variational principle and minimal points in product spaces, Nonlinear Anal. 39 (2000) 909–922; C. Tammer, A variational principle and a fixed point theorem, in: System Modelling and Optimization (Compiegne, 1993), in: Lecture Notes in Control and Inform. Sci., vol. 197, Springer, London, 1994, pp. 248–257].